The alliance on Monday urged Pakistan to re-open the supply line, escalating a row over the vital route.

Pakistan closed the main supply line across its territory following a US air strike last November that mistakenly killed 24 Pakistani soldiers.

Officials have since infuriated Nato by saying they will reopen the line, but only if they are paid the equivalent of £3,000 for every container or vehicle.

“Of course, it is frustrating,” Mr Cameron said, pointing out that Pakistan is nominally a “strong ally” of Britain and the West. He said he was “confident” that the routes will be reopened, but accepted that would not come soon.

“I'm confident that is the case from the discussions I have had. But it's not going to happen today,” he said.

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Supply lines are becoming increasingly important to the alliance as it prepares to leave Afghanistan. The US and its allies have billions of pounds worth of heavy equipment and vehicles in the country that can only be moved by road.

A Nato summit in Chicago issued a statement telling Pakistan it has an “important role in ensuring enduring peace, stability and security in Afghanistan and in facilitating the completion of the transition process.”

Nato will work with Pakistan “to reopen the ground lines of communication as soon as possible.”

The closure of the southern route has forced Nato leaders to try to open a supply corridor to the north-west through the former Soviet republics of Central Asia.

Philip Hammond, the defence secretary, earlier this year visited Uzbekistan to negotiate a supply route with its despotic leader, Islam Karimov.

British officials said the central Asian route is longer and more expensive to use than the southern one,

“We need one of these routes open and functioning to get our equipment out. Pakistan is the most obvious route,” said an official.